In this paper we develop a penalty method to approximate solutions of the free boundary problem for minimal surfaces. To this end we study the problem of finding minimizers of a functional Fλ which is defined as the sum of the Dirichlet integral and an appropriate penalty term weighted by a parameter λ. We prove existence of a solution for λ large enough as well as convergence to a solution of the free boundary problem as λ tends to infinity. Additionally regularity at the boundary of these solutions is shown, which is crucial for deriving numerical error estimates. Since every solution is harmonic, the analysis may be largely simplified by considering boundary values only and using harmonic extensions. In a subsequent paper we develop a fully discrete finite element procedure for approximating solutions to this one-dimensional problem and prove an error estimate which includes an order of convergence with respect to the grid size.

中文翻译：

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具有自由边界的最小曲面的近似

在本文中，我们开发了一种惩罚方法来逼近最小曲面的自由边界问题的解。为此，我们研究了寻找函数 Fλ 的最小值的问题，函数 Fλ 被定义为狄利克雷积分和由参数 λ 加权的适当惩罚项的总和。我们证明了 λ 的解存在足够大，并且随着 λ 趋于无穷大，自由边界问题的解的收敛性也存在。此外还显示了这些解决方案边界处的规律性，这对于推导数值误差估计至关重要。由于每个解都是调和的，因此可以通过仅考虑边界值和使用调和扩展来大大简化分析。